Symmetry

In the previous section, we found
when is
real. This fact is of high practical importance. It says that the
spectrum of every real signal is Hermitian.
Due to this symmetry, we may discard all negative-frequency spectral
samples of a real signal and regenerate them later if needed from the
positive-frequency samples. Also, spectral plots of real signals are
normally displayed only for positive frequencies; e.g., spectra of
sampled signals are normally plotted over the range 0 Hz to
Hz. On the other hand, the spectrum of a complex signal must
be shown, in general, from to (or from 0 to ),
since the positive and negative frequency components of a complex
signal are independent.

Recall from §7.3 that a signal is said to be
even if
, and odd if
. Below
are are Fourier theorems pertaining to even and odd signals and/or
spectra.

Theorem: If
, then
re is even and
im is odd.

Proof: This follows immediately from the conjugate symmetry of for real signals
.

Theorem: If
,
is even and is odd.

Proof: This follows immediately from the conjugate symmetry of expressed
in polar form
.

The conjugate symmetry of spectra of real signals is perhaps the most
important symmetry theorem. However, there are a couple more we can readily
show:

Theorem: An even signal has an even transform:

Proof:
Express in terms of its real and imaginary parts by
. Note that for a complex signal to be even, both its real and
imaginary parts must be even. Then

(7.5)

Let
even denote a function that is even in , such as
, and let
odd denote a function that is odd in
, such as , Similarly, let
even denote a
function of and that is even in both and , such as
, and
odd mean odd in both and .
Then appropriately labeling each term in the last formula above gives

Theorem: A real even signal has a real even transform:

(7.6)

Proof: This follows immediately from setting in the preceding
proof. From Eq.(7.5), we are left with

Definition: A signal with a real spectrum (such as any real, even signal)
is often called a zero phase signal. However, note that when
the spectrum goes negative (which it can), the phase is really
, not 0. When a real spectrum is positive at dc (i.e.,
), it is then truly zero-phase over at least some band
containing dc (up to the first zero-crossing in frequency). When the
phase switches between 0 and at the zero-crossings of the
(real) spectrum, the spectrum oscillates between being zero phase and
``constant phase''. We can say that all real spectra are
piecewise constant-phase spectra, where the two constant values
are 0 and (or , which is the same phase as ). In
practice, such zero-crossings typically occur at low magnitude, such
as in the ``side-lobes'' of the DTFT of a ``zero-centered symmetric
window'' used for spectrum analysis (see Chapter 8 and Book IV
[70]).